This work presents a unified framework for the unsupervised prediction of physically plausible interpolations between two 3D articulated shapes and the automatic estimation of dense correspondence between them. Interpolation is modelled as a diffeomorphic transformation using a smooth, time-varying flow field governed by Neural Ordinary Differential Equations (ODEs). This ensures topological consistency and non-intersecting trajectories while accommodating hard constraints, such as volume preservation, and soft constraints, e.g. physical priors. Correspondence is recovered using an efficient Varifold formulation, that is effective on high-fidelity surfaces with differing parameterisations. By providing a simple skeleton for the source shape only, we impose physically motivated constraints on the deformation field and resolve symmetric ambiguities. This is achieved without relying on skinning weights or any prior knowledge of the skeleton's target pose configuration. Qualitative and quantitative results demonstrate competitive or superior performance over existing state-of-the-art approaches in both shape correspondence and interpolation tasks across standard datasets.

We present an unsupervised framework for physically plausible shape interpolation and dense correspondence estimation between 3D articulated shapes. Our approach intentionally focuses upon pose variation within the same identity, which we believe is a meaningful and challenging problem in its own right. Our method uses Neural Ordinary Differential Equations (NODEs) to generate smooth flow fields that define diffeomorphic transformations, ensuring topological consistency and preventing self-intersections while accommodating hard constraints, such as volume preservation. By incorporating a lightweight skeletal structure, we impose kinematic constraints that resolve symmetries without requiring manual skinning or predefined poses. We enhance physical realism by interpolating skeletal motion with dual quaternions and applying constrained optimisation to align the flow field with the skeleton, preserving local rigidity. Additionally, we employ an efficient formulation of Normal Cycles, a metric from geometric measure theory, to capture higher-order surface details like curvature, enabling precise alignment between complex articulated structures and recovery of accurate dense correspondence mapping. Evaluations on multiple benchmarks show notable improvements over state-of-the-art methods in both interpolation quality and correspondence accuracy, with consistent performance across different skeletal configurations, demonstrating broad utility for shape matching and animation tasks.

Shape models have been used extensively to regularise segmentation of objects of interest in images, e.g. bones in medical x-ray radiographs, given supervised training examples. However, approaches usually adopt simple linear models that do not capture uncertainty and require extensive annotation effort to label a large number of set template landmarks for training. Conversely, supervised deep learning methods have been used on appearance directly (no explicit shape modelling) but these fail to capture detailed features that are clinically important.We present a supervised approach that combines both a non-linear generative shape model and a discriminative appearance-based convolutional neural network whilst quantifying uncertainty and relaxes the need for detailed, template based alignment for the training data. Our Bayesian framework couples the uncertainty from both the generator and the discriminator; our main contribution is the marginalisation of an intractable integral through the use of radial basis function approximations. We illustrate this model on the problem of segmenting bones from Psoriatic Arthritis hand radiographs and demonstrate that we can accurately measure the clinically important joint space gap between neighbouring bones.

Registration of images parameterized by landmarks provides a useful method of describing shape variations by computing the minimum-energy time-dependent deformation field that flows from one landmark set to the other. This is sometimes known as the geodesic interpolating spline and can be solved via a Hamiltonian boundary-value problem to give a diffeomorphic registration between images. However, small changes in the positions of the landmarks can produce large changes in the resulting diffeomorphism. We formulate a Langevin equation for looking at small random perturbations of this registration. The Langevin equation and three computationally convenient approximations are introduced and used as prior distributions. A Bayesian framework is then used to compute a posterior distribution for the registration, and also to formulate an average of multiple sets of landmarks.